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OpenCourseWare
Digital CommunicationsTelecommunications Engineering
Chapter 3
Angle modulations
Marcelino Lazaro
Departamento de Teora de la Senal y Comunicaciones
Universidad Carlos III de Madrid
Creative Commons License
1 / 5 1
Index of contents
Phase modulations (linear)I Phase shift keying (PSK) modulationsI Quadrature phase shift keying (QPSK) modulationI Offset quadrature phase shift keying (OQPSK) modulationI Differential PSK modulations
Non-linear modulationsI Frequency shift keying (FSK) modulationI Minimum shift keying (MSK) modulationI Continuous phase modulation (CPM)
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Spectrum of OQPSK modulation
Definitions
xI(t) =
2sI(t)cos(ct), xQ(t) =
2sQ(t)sin(ct)
Spectrum for each component (sk, k {I, Q})
Sxk(j) = 1
2
Ssk(j jc) +Ssk(j jc)
SsI(j) =
E{Re{A[n]}}
T|G(j)|2 , SsQ (j) =
E{Im{A[n]}}
T|G(j)|2
Spectrum of OQPSK modulation
Sx(j) = Es
2T
h|G(j jc)|2 +|G(j jc)|2
i
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Receivers for PSK modulations
- n@@6
ejct
y(t) -v(t) q
2T
R(n+1)TnT
dt -q[n]
Detector -A[n]
-y(t)
- n@@?
cos(ct)
- n@@6 sin(ct)
- q
2T
R(n+1)TnT
dt
- q2T R(n+1)T
nT dt
-Re{q[n]}
-Im{q[n]}
-q[n] Detector
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Receiver for OQPSK
TheT/2delay in the quadrature component is taken intoaccount (delay in the correlator)
-y(t)
- n@@?
cos(ct)
- n@@6
sin(ct)
- q
2T
R(n+1)TnT
dt
- q
2T
R(n+1)T+T/2
nT+T/2 dt
-Re{q[n]}
-Im{q[n]}
-q[n] Detector
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Effect of non-coherent receiver in PSK modulations
In a non-coherent receiver, phase of carriers used atreceiver to demodulate is different from phase of carriersused at the transmitter to modulate
I Difference of radians
The effect of this phase difference is that received
constellation is rotated radians
s
ss
s. .................... ...................
.................
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...
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...
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.................................
................... ....................
R-
a0a1
a2 a3
cs
cscs
c s . .................... ...................
.................
................
..................
.................
.
..................
........
........
...
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...
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................... ....................
R-.
....................................................................................
.............................................................................................
a0
a1
a2 a3
I
This effect can seriously affect performanceI However, non-coherent receivers have a lower costF Differential PSK modulation allows the use of non-coherent
receiverscMarcelino Lazaro, 2013 Digital Communications Nonlinear modulations 14/ 51
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Differential phase modulations
They do not require a coherent demodulation
PSK with differential phase enconding
[n] =[n 1] +[n]
-Bb[`] Encoder - m6
[n] -[n] Es ej()
z1
-A[n] g(t) -h62 ejct
s(t).
....................
................... -x(t)
Encoder forM-ary modulation
[n] 0,2
M, ,
2(M 1)M
Bit assignment is performed through [n]
Example: 4-PSK [n] 0
2
32
Bits 00 10 11 01 (Gray encoding)
Initialization: selection of an arbitrary (known) value for [1]I No error propagation because of initialization
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Differential PSK demodulators
-@@6
ejc t
y(t)-
v(t) 2f(t)
q(t)
?
q[n]
t= nT
-
@@6
ej
q[n]-
q0[n] PSK
Det.
-A[n]
Phase -
6- z1
-[n]
Coherent Receiver
-@@6
q[n]
- z
1 - () q[n
1]
- Phase - Detector -[n]
Non-coherent Receiver
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Non-coherent DPSK demodulator
Observation
q[n] =p
Esej([n]+) +z[n]
q[n
1] = pEs ej([n1]+) +z[n 1]Multiplier
q[n]q[n 1] =Esej([n][n1]) +p
Esej([n]+) z[n 1]
+p
Esej([n1]+) z[n] +z[n]z[n 1]
Detector
[n] = {q[n]q[n
1]}
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Probability of error for DPSK
Probability of error using coherent receivers
Pe 2PPSKeAn erroneous decided symbolA[n]affects two increments [n]and [n+1]
Probability of error with non-coherent receivers
I Statistic used for detection
q[n]q[n 1]Es
=p
Es ej([n][n1])
+ej([n]+) z[n 1]
+ej([n1]+) z[n] +z[n]z[n 1]
Es
I Three terms of noiseF Last one can be negligible for high Es/2zF The other two terms: independent, circularly symmetric
I Signal to noise ratio: 3 dB loss (double noise power)F Signal: EsF Noise:22z
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Orthogonality of CPFSK
Inner product of two pulses
hgi(t), g`(t)i =Z T
0
sin(it)sin(`t)dt
=1
2Z T
0cos((i `)| {z }
(NiN )2
T
t)dt1
2Z T
0cos((i+ `)| {z }
(Ni+N )2
T
t) dt
=T
2 [i `]
Pulses of CPFSK are orthogonal
Definition of an orthonormal base of dimension M
i(t) = r2T sin(it)wT(t), i= 0, 1, ,M 1CPFKS signal as an expansion in the orthonormal base
x(t) =p
EsX
n
A[n](t nT)
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FSK spectrum
Mean of the signal is periodic
Discrete spectrum (spectrum of the periodic mean)
SXd(j) = 2Es
T
1
(MT)2
M1
Xi=0Gi(j)
2
Xk
2k
T
Continuous spectrum (spectrum of the signal without the mean)
SXc(j) = 2Es
T
1
MT
M1Xi=0
|Gi(j)|2 1
M
M1X
i=0
Gi(j)
2
FSK - Power spectral density
SX(j) =SXc(j) +SXd(j)
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Receivers for FSK modulation
Coherent receiver with matched filters or correlators
Pe =Q
rEs
N0
Efect of phase error - Example:n = 0,A[n] =i, phase error
y(t) =
r2Es
Tsin(it+)wT(t)
q`[0] =
Z T0
y(t)`(t) dt=
Z T0
r2Es
Tsin(it+)
r2
Tsin(`t) dt
=
Es
T Z T
0
[cos((i
`)t+)
cos(i+`)t+)] dt
=p
Escos()[i `]
I Atenuation term:cos()
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Coherent receiver for binary FSK
y(t)
- 0(t)
- 1(t)
?
-q0[n]
-q1[n]
t= nT
Maximum -B[n]
Coherent Receiver
y(t)
- h0(t)
- h1(t)
- Envelopedetector
- Envelopedetector
?
-q0[n]
-q1[n]
t= nT
Maximum -B[n]
Non-coherent Receiver
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Non-coherent (quadratic law) receiver for FSK
-y(t)
-
-@@6
sin(0t)
-@@?
cos(0t)
-R(n+1)TnT
dt - ()2
-R(n+1)TnT
dt - ()2?
6
-
r0[n]
-
-@@6
sin(1t)
-@@?
cos(1t)
-R(n+1)TnT
dt - ()2
-
R(n+1)T
nT dt - ()2
?
6
-r1[n]
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FSK seem as frequency shift from a central frequency
Definition of central frequency
c=0+M1
2 =
T C, C Z, Codd
I Value of central frequency: c=
T odd integer
I Frecuencies of the pulse for symbol of discrete index n
c+I[n]
T
Encoder
I[n] {1,3, ,(M 1)}FSK analytic expression as shift from c
x(t) = r2EsT
Xn
sinct+I[n] tT wT(t nT)
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Minimum shift keying (MSK) modulation
Information: discrete frequency changes in the frequency of a carrier
Orthogonality of carriers with minimum frequency separation
Inner product of pulses gi(t)
hgi,g`i =Z T
0
sin(it)sin(`t) dt
=1
2
Z T0
cos[(i `)t]dt 12
cos[(i+`)t]dt
=1
2
sin[(i `)T](i `)
1
2
sin[(i+`)T]
(i+`)
Minimum required frequency separation (in narrow band systems)
I Assumption: sin[(i+`)T]
(i+`) can be neglected (high denominator)
i `= TNi,`, i,j= 0, 1, ,M 1, i 6=`
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Minimum shift keying (MSK) modulation (II)
Key differences with CPFSK modulation
I Separation between consecutive frequencies is half for MSKF MSK: =i i1 = TF CPFSK: =i i1 = 2T
I Values fori are not constrained to be integer multiples of 2
T as in
CPFSK (neither integer multiples of T
)F Frequency selection does not automatically provides phase continuity
F Memory must be introduced to provide phase continuity
MSK signal using central frequency notation
x(t) =
r2Es
TX
n
sinct+I[n]
t
2T+[n]
wT(t nT)
I Encoder: I[n] {1,3, ,(M 1)}I Phase continuity is achieved by introducing memory term [n]
[n] =[n 1] + n2 (I[n 1] I[n]) , mod2
Recursive estimation of accumulated phase at the end of symbol intervals
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MSK spectrumAlternative expression for MSK
x(t) =p
2Escos(ct)X
even n
I[n] cos([n])(1)n/2g(t nT)
+ p2Essin(ct) Xeven n cos([n])(1)n/2g(t nT+ T)Similar to OQPSK
I Modified symbolsI Pulse:
g(t) = r1
Tsin
t
2Tw2T(t), |G(j)|2 =16T2
cos(T)
2
42T2
2
MSK spectrum
SX(j) =8Es2
cos[( c)T]2 4( c)2T2
2+8Es
2
cos[(+c)T]
2 4(+c)2T22
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Receiver for MSK modulation
Demodulator based on the ML receiver for FSK
Demodulator based on the ML receiver for OQPSK
Probability of error
Pe=2QrEsN0I Memory is not taken into account at the receiverI Optimum receiver has a higher complexity
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Receiver for binary MSK
Sub-optimal MSK binary receiver based on a FSK receiver
where the absolute value evaluation for each possible
frequency is introduced to consider different possible initial
phases
y(t)
- 0(t)
- 1(t)
?
-q0[n] abs()
-q1[n] abs()
t= nT
Maximum -B[n]
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Continuous phase (CPM) modulations
Family of modulations including CPFSK and MSK modulations
Basic characteristics
I Constant envelopeI Phase continuityI Bandwith reduction: smoothing the evolution of the instantaneous
phase
CPM signal: analytic expression in the time domain
x(t) =
r2Es
Tsin [ct+0+(t,I)]
I I: Sequence of transmitted symbolsI c: nominal carrier frequencyI 0: initial carrier phaseI Es: mean energy transmitted in a symbol period
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Generation of the CPM signal
Encoder:I[n] {1,3, ,(M 1)}Base band PAM signal
s(t) = Xn
I[n]g(t
nT)
Pulseg(t)is causal, of length T, and normalized
Normalization:
Z
g(t)dt= 1
2
CPM signal: instantaneous frequencyc+2d T s(t)
Instantaneous phase is obtained by integrating this frequency
(t,I) =2d T
Z t
s()d
I d: peak frequency deviation
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Time domain expression for CPM
x(t) =
r2Es
Tsin
ct+ 0+
(t,I)z }| {2 d T
Z t
Xn
I[n] g( nT)d| {z }
s()
Phase value(t,I)inside interval[nT, (n+1)T](symbol interval forI[n])
(t,I) =2d T Z t
s() d=[n] +(t, n)
I [n]: phase that has been accumulated up to t= nT:F Due to previous transmitted symbols (up toI[n 1])
[n] =d Tn1X
m=
I[m]
I (t, n): incremental phase starting fromt= nT:F Due only to current symbol I[n]
(t, n) =2d T I[n]qg(t nT), beingqg(t) =Z t
g()d
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Time domain expression for CPM - Modulation index
Alternative time domain expression introducing a different
parameter (replacing peak frequency deviation)
Definition of modulation index h:
h= d T
Phase value in the symbol interval associated toI[n]:I [n]: accumulated phase up to t= nT:
[n] = hn1X
m=I[m]
I (t, n): incremental phase fromt= nT:
(t, n) =2 hI[n]qg(t nT)
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Identification of binary CPFSK modulation as a CPM
Analytic expression for a CPFSK modulation
x(t) =
r2Es
TX
n
sinct+I[n]
t
T
wT(t nT)
Binary CPFSK as a CPM:d=
T, h= 1
Considering[0] =0
(t,I) =n1Xm=0
I[m] +2I[n](t nT)
2T=
n1Xm=0
I[m] nI[n] + tT
I[n]
I Taking into account that
n1Xm=0
I[m] n I[n] =K 2, K Z
I Phase(t,I)is, 2 modulus
(t,I) = t
TI[n] =
t
T
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Identification of MSK modulation as a CPM
MSK signal in the time domain
x(t) = r2Es
T Xnsinct+I[n]
t
2T+[n] wT(t nT)
Parameters identifying MSK as a CPM
d=
2T, h=
1
2
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Phase trees in CPM modulations
Drawing of possible phase evolution in time starting from an initial phase
Transitions in a symbol interval
I Phase increment in each symbol interval
((n+1)T) (nT) =[n+1] [n] = hI[n]
I Shape for moving from the value of phase at the begining of
symbol interval to the value of phase at the end of symbol interval
F Proportional to the integral of pulseg(t), i.e.,qg(t)
(t, n) =2hI[n]qg(t nT)
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Pulses for partial response CPM
Raised cosine pulses
g(t) = 1
2LT1 cos
2t
LTwLT(t)
I Smoothing phase transitions
Gaussian MSK (GMSK)
g(t) = 1
2T
Q
2(t T/2)
ln 2
Q
2(t+T/2)
ln 2
I Employed in GSM (=0,3) and DECT (=0,2)I Squared pulse filtered with a Gaussian impulse response
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Phase tree - Partial response CPM - Example
Example: raised cosine pulse (L= 2)
g(t) = 1
4T
1 cos
2t
2T
w2T(t), qg(t) =
0, t< 01
4T
t 2T
2sin
2t2T
0 t